Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{k^2 - 64}{k + 8}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = k$ $ b = \sqrt{64} = 8$ So we can rewrite the expression as: $r = \dfrac{({k} + {8})({k} {-8})} {k + 8} $ We can divide the numerator and denominator by $(k + 8)$ on condition that $k \neq -8$ Therefore $r = k - 8; k \neq -8$